Chance Encounters

1: The Educational Perspective.- 1. Aims and Rationale.- 2. Views on Didactics.- Fischer's open mathematics.- Fischbein's interplay between intuitions and mathematics.- Freudenthal's didactical phenomenology.- Bauersfeld's subjective domains of experience.- 3. Basic Ideas of the Chapters.- A probabilistic perspective.- Empirical research in understanding probability.- Analysis of the probability curriculum.- The theoretical nature of probability in the classroom.- Computers in probability education.- Psychological research in probabilistic understanding.- 2: A Probabilistic Perspective.- 1. History and Philosophy.- Tardy conceptualization of probability.- The rule of ' favourable to possible'.- Expectation and frequentist applications.- Inference and the normal law.- Foundations and obstacles.- Axiomatization of probability.- Modern views on probability.- 2. The Mathematical Background.- Model-building.- Assigning probabilities.- Conditional probability.- Random variables and distributions.- Central theorems.- Standard situations.- 3. Paradoxes and Fallacies.- Chance assignment.- Expectation.- Independence and dependence.- Logical curiosities.- Concluding comments.- 3: Empirical Research in Understanding Probability.- 1. Research Framework.- Peculiarities of stochastics and its teaching.- Research in psychology and didactics.- 2. Sample Space and Symmetry View.- Nod : Tossing a counter.- No.2: Hat lottery.- 3. Frequentist Interpretation.- No.3: The six children.- No.4: Snowfall.- 4. Independence and Dependence.- No.5: Dependent urns.- No.6: Independent urns.- 5. Statistical Inference.- No.7: Coin tossing.- No.8: Drawing from a bag.- 6. Concluding Comments.- Empirical research.- Teaching consequences.- 4: Analysis of the Probability Curriculum.- 1. General Aims.- Objectives.- Ideas.- Skills.- Inclination to apply ideas and skills.- 2. General Curriculum Issues.- Aspects of the curriculum.- Curriculum sources.- Choice of orientation.- 3. Curriculum Issues in Probability.- Student readiness.- Different approaches to probability curriculum.- 4. Approaches to the Probability Curriculum.- What to look for?.- Research needs.- 5: The Theoretical Nature of Probability in the Classroom.- 1. Approaches towards Teaching.- Structural approaches.- 2. The Theoretical Nature of Stochastic Knowledge.- Approaches to teaching probability.- Theoretical nature of probability.- Objects, signs and concepts.- 3. Didactic Means to Respect the Theoretical Nature of Probability.- Interrelations between mathematics and exemplary applications.- Means of representation and activities.- 4. On the Didactic Organization of Teaching Processes.- The role of teachers.- The role of task systems.- 5. Discussion of an Exemplary Task.- Didactic framework of the task.- Classroom observations.- Implications for task systems.- 6: Computers in Probability Education.- 1. Computers and Current Practice in Probability Teaching.- Pedagogical problems and perspectives.- Changes in probability, statistics, and in their applications.- Changing technology and its influence on pedagogical ideas.- 2. Computers as Mathematical Utilities.- The birthday problem.- Exploring Bayes' formula.- Binomial probabilities.- Programming languages and other tools.- 3. Simulation as a Problem Solving Method.- Integrating simulation, algorithmics and programming.- Simulation as an alternative to solving problems analytically.- The potential of computer-aided simulation.- Software for simulation and modelling.- Computer generated random numbers.- 4. Simulation and Data Analysis for Providing an Empirical Background for Probability.- Making theoretical objects experiential.- Beginning with' limited' technological equipment.- Laws of large numbers and frequentist interpretation.- Random sampling and sampling variation.- Structure in random sequences.- A simulation and modelling tool as companion of the curriculum.- Games and strategy.- 5. Visualization, Graphical Methods and Animation.- 6. Concluding Remarks.- Software/Bibliography.- 7: Psychological Research in Probabilistic Understanding.- 1. Traditional Research Paradigms.- Probability learning.- Bayesian revision.- Disjunctive and conjunctive probabilities.- Correlation.- 2. Current Research Paradigms.- Judgemental heuristics.- Structure and process models of thinking.- Probability calibration.- Event-related brain potential research.- Overview on research paradigms.- 3. Critical Dimensions of Educational Relevance.- The conception of the task.- The conception of the subject.- The conception of the subject-task relation.- 4. Developmental Approaches on the Acquisition of the Probability Concept.- The cognitive-developmental approach of Piaget and Inhelder.- Fischbein's learning-developmental approach.- Information processing approaches.- Semantic-conceptual and operative knowledge approach.- Discussion of the developmental approaches.- Looking Forward.